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Use the new lemmas

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aisejohan committed Nov 27, 2019
1 parent f18336a commit 1bdb23d651052e03ad682da755ee8ecdfc95a50e
Showing with 41 additions and 3 deletions.
  1. +1 −1 discriminant.tex
  2. +39 −1 more-morphisms.tex
  3. +1 −1 tags/tags
@@ -2637,7 +2637,7 @@ \section{The Tate map}
Lb^*\NL_{Y/X} \longrightarrow \NL_{Y'/X'}
$$
is a quasi-isomorphism by
More on Morphisms, Lemma \ref{more-morphisms-lemma-base-change-NL}.
More on Morphisms, Lemma \ref{more-morphisms-lemma-base-change-NL-flat}.
Thus we get a canonical isomorphism
$b^*\det(\NL_{Y/X}) \to \det(\NL_{Y'/X'})$ which sends the
canonical section $\delta(\NL_{Y/X})$ to $\delta(\NL_{Y'/ X'})$, see
@@ -3824,6 +3824,25 @@ \section{The naive cotangent complex}
Special case of Modules, Lemma \ref{modules-lemma-exact-sequence-NL}.
\end{proof}

\begin{lemma}
\label{lemma-base-change-NL}
Consider a cartesian diagram of schemes
$$
\xymatrix{
X' \ar[r]_{g'} \ar[d] & X \ar[d] \\
Y' \ar[r] & Y
}
$$
The canonical map $(g')^*\NL_{X/Y} \to \NL_{X'/Y'}$ induces
an isomorphism on $H^0$ and a surjection on $H^{-1}$.
\end{lemma}

\begin{proof}
Translated into algebra this is
More on Algebra, Lemma \ref{more-algebra-lemma-base-change-NL}.
To do the translation use Lemma \ref{lemma-NL-affine}.
\end{proof}

\begin{lemma}
\label{lemma-flat-base-change-NL}
Consider a cartesian diagram of schemes
@@ -3843,7 +3862,7 @@ \section{The naive cotangent complex}
\end{proof}

\begin{lemma}
\label{lemma-base-change-NL}
\label{lemma-base-change-NL-flat}
Consider a cartesian diagram of schemes
$$
\xymatrix{
Divided Power Algebra, Proposition \ref{dpa-proposition-regular-ideal}.
\end{proof}

\begin{lemma}
\label{lemma-lci-NL}
Let $f : X \to Y$ be a local complete intersection homomorphism.
Then the naive cotangent complex $\NL_{X/Y}$ is a perfect object
of $D(\mathcal{O}_X)$ of tor-amplitude in $[-1, 0]$.
\end{lemma}

\begin{proof}
Translated into algebra this is
More on Algebra, Lemma \ref{more-algebra-lemma-lci-NL}.
To do the translation use
Lemmas \ref{lemma-affine-lci} and
\ref{lemma-NL-affine} as well as
Derived Categories of Schemes, Lemmas
\ref{perfect-lemma-affine-compare-bounded},
\ref{perfect-lemma-tor-dimension-affine} and
\ref{perfect-lemma-perfect-affine}.
\end{proof}

\begin{lemma}
\label{lemma-perfect-NL-lci}
Let $f : X \to Y$ be a perfect morphism of locally Noetherian schemes.
0FJX,perfect-lemma-determinant-two-term-complexes
0FJY,perfect-remark-functorial-det
0FJZ,more-morphisms-lemma-flat-base-change-NL
0FK0,more-morphisms-lemma-base-change-NL
0FK0,more-morphisms-lemma-base-change-NL-flat
0FK1,more-morphisms-lemma-perfect-conormal-free-lci
0FK2,more-morphisms-lemma-perfect-NL-lci
0FK3,more-morphisms-lemma-flat-fp-NL-lci

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